Building a Maclaurin series
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Flip to reveal answersWhat is the coefficient of xⁿ in a Maclaurin series?
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All 8 Flashcards — Building a Maclaurin series
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Question
What is the coefficient of xⁿ in a Maclaurin series?
Answer
f⁽ⁿ⁾(0) ÷ n! — the nth derivative evaluated at x = 0, divided by n!.
Question
Write the general Maclaurin series of f(x).
Answer
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …
Question
Why does the Maclaurin formula divide by n!?
Answer
Differentiating xⁿ exactly n times gives n!; dividing by n! makes the nth term contribute exactly f⁽ⁿ⁾(0) to the nth derivative at 0.
Question
Maclaurin series of eˣ?
Answer
eˣ = 1 + x + x²/2! + x³/3! + … (every power, denominator n!).
Question
Maclaurin series of sin x?
Answer
sin x = x − x³/3! + x⁵/5! − … (odd powers only, alternating signs).
Question
Maclaurin series of cos x?
Answer
cos x = 1 − x²/2! + x⁴/4! − … (even powers only, alternating signs).
Question
Maclaurin series of ln(1 + x)?
Answer
ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … (plain denominators, alternating signs).
Question
Why does sin x contain only odd powers?
Answer
Every even derivative of sin x equals ±sin 0 = 0, so all even-power coefficients vanish.
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Topic 5.19 hub
Maclaurin series (HL only)
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